where we assume the eigenvalues to be ordered in decreasing order so that \({\cal V}({\bf A})\)  is non
negative. Moreover, for any constant \(c\) , we have \({\cal V}(c{\bf A}) = c^{m(m-1)/2}{\cal V}({\bf A})\) . Let also
\({\bf F}=\lbrace f_i(a_j)\rbrace \) , \(i,j=1,\ldots ,m\) , be an \(m\times m\)  matrix, where \(f_i(\cdot )\) 's are any
differentiable functions. Clearly, if the eigenvalues of \({\bf A}\)  are not distinct, \({\cal V}({\bf A})=0\) 
and \(|{\bf F}|=0\) . In such a case, the ratio \(|{\bf F}|/{\cal V}({\bf A})\) , which appears in the density of
many matrices that we study in the following, can be evaluated by applying l'Hôpital's rule.
More precisely, let \(n\)  be an integer such that \(0 < n < m\) , then [1]}
\(\lim _{a_{n+1},\ldots ,a_m \rightarrow a}\frac{|{\bf F}|}{{\cal V}({\bf A})} =\frac{\pi _m\Gamma _n(m)}{\pi _n\Gamma _m(m)} \frac{|\widetilde{{\bf F}}|}{{\cal V}(\widetilde{{\bf A}})}|\widetilde{{\bf A}} -a{\bf I}|^{n-m}\) 
